3.1569 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=300 \[ \frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^3}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^6 (a+b x) (d+e x)^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^5}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^2} \]

[Out]

((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^6*(a + b*x)*(d + e*x)^5) - (5
*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^6*(a + b*x)*(d + e*x)^4) +
(10*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d + e*x)^
3) - (5*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d + e*x
)^2) + (5*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d + e*x
)) + (b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^6*(a + b*x))

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Rubi [A]  time = 0.404783, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^3}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^6 (a+b x) (d+e x)^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^6 (a+b x) (d+e x)^5}+\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) (d+e x)}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^6,x]

[Out]

((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^6*(a + b*x)*(d + e*x)^5) - (5
*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^6*(a + b*x)*(d + e*x)^4) +
(10*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d + e*x)^
3) - (5*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d + e*x
)^2) + (5*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d + e*x
)) + (b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^6*(a + b*x))

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Rubi in Sympy [A]  time = 46.2542, size = 235, normalized size = 0.78 \[ \frac{b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{6} \left (a + b x\right )} - \frac{b^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{6} \left (a + b x\right ) \left (d + e x\right )} - \frac{b^{3} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6 e^{4} \left (d + e x\right )^{2}} - \frac{b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{3} \left (d + e x\right )^{3}} - \frac{b \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{20 e^{2} \left (d + e x\right )^{4}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 e \left (d + e x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**6,x)

[Out]

b**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(d + e*x)/(e**6*(a + b*x)) - b**4*(a*e
- b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(e**6*(a + b*x)*(d + e*x)) - b**3*(3*a +
3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(6*e**4*(d + e*x)**2) - b**2*(a**2 + 2*
a*b*x + b**2*x**2)**(3/2)/(3*e**3*(d + e*x)**3) - b*(5*a + 5*b*x)*(a**2 + 2*a*b*
x + b**2*x**2)**(3/2)/(20*e**2*(d + e*x)**4) - (a**2 + 2*a*b*x + b**2*x**2)**(5/
2)/(5*e*(d + e*x)**5)

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Mathematica [A]  time = 0.322763, size = 196, normalized size = 0.65 \[ \frac{\sqrt{(a+b x)^2} \left ((b d-a e) \left (12 a^4 e^4+3 a^3 b e^3 (9 d+25 e x)+a^2 b^2 e^2 \left (47 d^2+175 d e x+200 e^2 x^2\right )+a b^3 e \left (77 d^3+325 d^2 e x+500 d e^2 x^2+300 e^3 x^3\right )+b^4 \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 b^5 (d+e x)^5 \log (d+e x)\right )}{60 e^6 (a+b x) (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^6,x]

[Out]

(Sqrt[(a + b*x)^2]*((b*d - a*e)*(12*a^4*e^4 + 3*a^3*b*e^3*(9*d + 25*e*x) + a^2*b
^2*e^2*(47*d^2 + 175*d*e*x + 200*e^2*x^2) + a*b^3*e*(77*d^3 + 325*d^2*e*x + 500*
d*e^2*x^2 + 300*e^3*x^3) + b^4*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d
*e^3*x^3 + 300*e^4*x^4)) + 60*b^5*(d + e*x)^5*Log[d + e*x]))/(60*e^6*(a + b*x)*(
d + e*x)^5)

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Maple [A]  time = 0.021, size = 383, normalized size = 1.3 \[{\frac{300\,\ln \left ( ex+d \right ){x}^{4}{b}^{5}d{e}^{4}+600\,\ln \left ( ex+d \right ){x}^{3}{b}^{5}{d}^{2}{e}^{3}+300\,\ln \left ( ex+d \right ) x{b}^{5}{d}^{4}e-30\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-12\,{a}^{5}{e}^{5}+137\,{b}^{5}{d}^{5}+600\,\ln \left ( ex+d \right ){x}^{2}{b}^{5}{d}^{3}{e}^{2}+60\,\ln \left ( ex+d \right ){x}^{5}{b}^{5}{e}^{5}+60\,\ln \left ( ex+d \right ){b}^{5}{d}^{5}-300\,{x}^{4}a{b}^{4}{e}^{5}+300\,{x}^{4}{b}^{5}d{e}^{4}-300\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+900\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}-200\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+1100\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}-75\,x{a}^{4}b{e}^{5}+625\,x{b}^{5}{d}^{4}e-15\,{a}^{4}bd{e}^{4}-20\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-60\,a{b}^{4}{d}^{4}e-600\,{x}^{3}a{b}^{4}d{e}^{4}-300\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-600\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-100\,x{a}^{3}{b}^{2}d{e}^{4}-150\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-300\,xa{b}^{4}{d}^{3}{e}^{2}}{60\, \left ( bx+a \right ) ^{5}{e}^{6} \left ( ex+d \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^6,x)

[Out]

1/60*((b*x+a)^2)^(5/2)*(300*ln(e*x+d)*x^4*b^5*d*e^4+600*ln(e*x+d)*x^3*b^5*d^2*e^
3+300*ln(e*x+d)*x*b^5*d^4*e-30*a^2*b^3*d^3*e^2-12*a^5*e^5+137*b^5*d^5+600*ln(e*x
+d)*x^2*b^5*d^3*e^2+60*ln(e*x+d)*x^5*b^5*e^5+60*ln(e*x+d)*b^5*d^5-300*x^4*a*b^4*
e^5+300*x^4*b^5*d*e^4-300*x^3*a^2*b^3*e^5+900*x^3*b^5*d^2*e^3-200*x^2*a^3*b^2*e^
5+1100*x^2*b^5*d^3*e^2-75*x*a^4*b*e^5+625*x*b^5*d^4*e-15*a^4*b*d*e^4-20*a^3*b^2*
d^2*e^3-60*a*b^4*d^4*e-600*x^3*a*b^4*d*e^4-300*x^2*a^2*b^3*d*e^4-600*x^2*a*b^4*d
^2*e^3-100*x*a^3*b^2*d*e^4-150*x*a^2*b^3*d^2*e^3-300*x*a*b^4*d^3*e^2)/(b*x+a)^5/
e^6/(e*x+d)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21281, size = 504, normalized size = 1.68 \[ \frac{137 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e - 30 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 15 \, a^{4} b d e^{4} - 12 \, a^{5} e^{5} + 300 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \,{\left (3 \, b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 100 \,{\left (11 \, b^{5} d^{3} e^{2} - 6 \, a b^{4} d^{2} e^{3} - 3 \, a^{2} b^{3} d e^{4} - 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \,{\left (25 \, b^{5} d^{4} e - 12 \, a b^{4} d^{3} e^{2} - 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - 3 \, a^{4} b e^{5}\right )} x + 60 \,{\left (b^{5} e^{5} x^{5} + 5 \, b^{5} d e^{4} x^{4} + 10 \, b^{5} d^{2} e^{3} x^{3} + 10 \, b^{5} d^{3} e^{2} x^{2} + 5 \, b^{5} d^{4} e x + b^{5} d^{5}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^6,x, algorithm="fricas")

[Out]

1/60*(137*b^5*d^5 - 60*a*b^4*d^4*e - 30*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 1
5*a^4*b*d*e^4 - 12*a^5*e^5 + 300*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 300*(3*b^5*d^2*e^
3 - 2*a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 100*(11*b^5*d^3*e^2 - 6*a*b^4*d^2*e^3 - 3
*a^2*b^3*d*e^4 - 2*a^3*b^2*e^5)*x^2 + 25*(25*b^5*d^4*e - 12*a*b^4*d^3*e^2 - 6*a^
2*b^3*d^2*e^3 - 4*a^3*b^2*d*e^4 - 3*a^4*b*e^5)*x + 60*(b^5*e^5*x^5 + 5*b^5*d*e^4
*x^4 + 10*b^5*d^2*e^3*x^3 + 10*b^5*d^3*e^2*x^2 + 5*b^5*d^4*e*x + b^5*d^5)*log(e*
x + d))/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x
+ d^5*e^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224182, size = 510, normalized size = 1.7 \[ b^{5} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ){\rm sign}\left (b x + a\right ) + \frac{{\left (300 \,{\left (b^{5} d e^{3}{\rm sign}\left (b x + a\right ) - a b^{4} e^{4}{\rm sign}\left (b x + a\right )\right )} x^{4} + 300 \,{\left (3 \, b^{5} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 2 \, a b^{4} d e^{3}{\rm sign}\left (b x + a\right ) - a^{2} b^{3} e^{4}{\rm sign}\left (b x + a\right )\right )} x^{3} + 100 \,{\left (11 \, b^{5} d^{3} e{\rm sign}\left (b x + a\right ) - 6 \, a b^{4} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 3 \, a^{2} b^{3} d e^{3}{\rm sign}\left (b x + a\right ) - 2 \, a^{3} b^{2} e^{4}{\rm sign}\left (b x + a\right )\right )} x^{2} + 25 \,{\left (25 \, b^{5} d^{4}{\rm sign}\left (b x + a\right ) - 12 \, a b^{4} d^{3} e{\rm sign}\left (b x + a\right ) - 6 \, a^{2} b^{3} d^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, a^{3} b^{2} d e^{3}{\rm sign}\left (b x + a\right ) - 3 \, a^{4} b e^{4}{\rm sign}\left (b x + a\right )\right )} x +{\left (137 \, b^{5} d^{5}{\rm sign}\left (b x + a\right ) - 60 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) - 30 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) - 20 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) - 15 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) - 12 \, a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^6,x, algorithm="giac")

[Out]

b^5*e^(-6)*ln(abs(x*e + d))*sign(b*x + a) + 1/60*(300*(b^5*d*e^3*sign(b*x + a) -
a*b^4*e^4*sign(b*x + a))*x^4 + 300*(3*b^5*d^2*e^2*sign(b*x + a) - 2*a*b^4*d*e^3
*sign(b*x + a) - a^2*b^3*e^4*sign(b*x + a))*x^3 + 100*(11*b^5*d^3*e*sign(b*x + a
) - 6*a*b^4*d^2*e^2*sign(b*x + a) - 3*a^2*b^3*d*e^3*sign(b*x + a) - 2*a^3*b^2*e^
4*sign(b*x + a))*x^2 + 25*(25*b^5*d^4*sign(b*x + a) - 12*a*b^4*d^3*e*sign(b*x +
a) - 6*a^2*b^3*d^2*e^2*sign(b*x + a) - 4*a^3*b^2*d*e^3*sign(b*x + a) - 3*a^4*b*e
^4*sign(b*x + a))*x + (137*b^5*d^5*sign(b*x + a) - 60*a*b^4*d^4*e*sign(b*x + a)
- 30*a^2*b^3*d^3*e^2*sign(b*x + a) - 20*a^3*b^2*d^2*e^3*sign(b*x + a) - 15*a^4*b
*d*e^4*sign(b*x + a) - 12*a^5*e^5*sign(b*x + a))*e^(-1))*e^(-5)/(x*e + d)^5